Hopf Algebras, Renormalization and Noncommutative Geometry

TL;DR

The paper links the Hopf algebra for renormalization in quantum field theory (rooted trees) to the Hopf algebra arising in noncommutative geometry’s transverse index theory (diffeomorphism-coordinates) by embedding both in a universal Hochschild-cohomology framework. It shows ${\cal H}_R$ is dual to the enveloping algebra of a Lie algebra ${\cal L}^1$ generated by rooted trees, and constructs a map $\Theta$ to the Lie algebra of formal vector fields, identifying ${\cal H}_R$ as coordinates on a nilpotent formal group mirroring ${\cal H}_T$. The antipode in ${\cal H}_R$ corresponds to renormalization counterterm extraction, with the rooted-tree structure capturing subdivergences and forest relations, while overlaps are shown to reduce to sums of trees (via the appendix and φ^3 theory example). Together these results provide a conceptual bridge between renormalization calculus and the refined calculus underlying transverse geometry in noncommutative geometry.

Abstract

We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of transverse index theory for foliations.

Figures (14)

• Figure 1: A toy model realizing rooted trees. We define the first couple of rooted trees $t_1,t_2,t_{3_1}, t_{3_2}$. The root is always drawn as the uppermost vertex. $t_2$ gives rise to the function $x_2(c)$.
• Figure 2: The action of $B_-$ on a rooted tree.
• Figure 3: The action of $B_+$ on a monomial of trees.
• Figure 4: An elementary cut splits a rooted tree $t$ into two components $t_1,t_2$.
• Figure 5: An admissible cut $C$ acting on a tree $t$. It produces a monomial of trees. One of the factors, $R^C(t)$, contains the root of $t$.